A holomorphic quadratic differential \(q\)  on a Riemannian surface \(X\)  is a holomorphic section of \(T^*X\otimes T^*X\) , where \(T^*X\)  is the holomorphic cotangent bundle. In a local coordinate \(z\) , \(q=f(z)\,dz^2\)  for some holomorphic function \(f(z)\) , see [1]}.
